Question

Write the equation in slope-intercept form and find the equation of a line through the given point (a) parallel and (b) perpendicular to the given line.
(2, 1) 4x - 2y = 3

Answer 1

keeping in mind that parallel lines have the same exact slope, hmmm what's the slope of 4x - 2y = 3 anyway?  well, let's put it in slope-intercept form firstly.

$\bf 4x-2y=3\implies 4x-3=2y\implies \cfrac{4x-3}{2}=y \\\\\\ \cfrac{4x}{2}-\cfrac{3}{2}=y\implies \stackrel{slope}{2}x-\cfrac{3}{2}=y$

a)

so we're looking for a line whose slope is 2 and passes through 2,1.

$\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{1})~\hspace{10em} slope = m\implies 2 \\\\\\ \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-1=2(x-2) \\\\\\ y-1=2x-4\implies \blacktriangleright y=2x-3 \blacktriangleleft$

b)

$\bf \stackrel{\textit{perpendicular lines have \underline{negative reciprocal} slopes}} {\stackrel{slope}{2\implies \cfrac{2}{1}}\qquad \qquad \qquad \stackrel{reciprocal}{\cfrac{1}{2}}\qquad \stackrel{negative~reciprocal}{-\cfrac{1}{2}}} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\bf (\stackrel{x_1}{2}~,~\stackrel{y_1}{1})~\hspace{10em} slope = m\implies -\cfrac{1}{2} \\\\\\ \stackrel{\textit{point-slope form}}{y- y_1= m(x- x_1)}\implies y-1=-\cfrac{1}{2}(x-2) \\\\\\ y-1=-\cfrac{1}{2}x+1\implies \blacktriangleright y=-\cfrac{1}{2}x+2 \blacktriangleleft$